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Cubic graphs, their Ehrhart quasi-polynomials, and a
scissors congruence phenomenon
Cristina Fernandes, José de Pina, Jorge Luis Ramírez Alfonsín, Sinai Robins
To cite this version:
Cubic graphs, their Ehrhart quasi-polynomials,
and a scissors congruence phenomenon
∗
Cristina G. Fernandes
†José C. de Pina
†Jorge Luis Ramírez Alfonsín
‡Sinai Robins
†February 20, 2018
Abstract
The scissors congruence conjecture for the unimodular group is an analogue of Hilbert’s third problem, for the equidecomposability of polytopes. Liu and Osser-man studied the Ehrhart quasi-polynomials of polytopes naturally associated to graphs whose vertices have degree one or three. In this paper, we prove the scissors congruence conjecture, posed by Haase and McAllister, for this class of polytopes. The key ingredient in the proofs is the nearest neighbor interchange on graphs and a naturally arising piecewise unimodular transformation.
1
Introduction
A cubic graph is a graph whose vertices have degree three and a {1, 3}-graph is a graph whose vertices have degree either one or three. The graphs are allowed to have loops and parallel edges. Motivated by a result of Mochizuki [11], Liu and Osserman [10] associated
a polytope PG to each {1, 3}-graph G and studied its Ehrhart quasi-polynomial.
For each degree three vertex v of a {1, 3}-graph G = (V, E), let a, b, and c be the three edges incident to v. Denote by S(v) the linear system consisting of a perimeter inequality and three metric inequalities defined on the variables wa, wb, and wc as follows:
wa+ wb+ wc ≤ 1
wa ≤ wb+ wc
wb ≤ wa+ wc
wc ≤ wa+ wb.
∗FAPESP 13/03447-6 and 15/10323-7, CNPq 456792/2014-7 and 452507/2016-2, CAPES PROEX. †Instituto de Matemática e Estatística, Universidade de São Paulo, 05508-090 São Paulo, Brazil.
(cris@ime.usp.br, coelho@ime.usp.br, srobins@ime.usp.br).
From the three metric inequalities, one can immediately conclude that wa, wb, and wc are nonnegative. Consider the union of all the linear systems S(v), taken over all degree
three vertices v of G. The polytope PG is defined by the set of all real solutions for this
linear system.
Let E = {1, . . . , m} and w : E → R be a weight function defined on the edges of G. We use the vector notation w = (w1, . . . , wm) ∈ Rm. In particular, when w is a solution for the linear system defining PG, we write w ∈ PG.
Given a rational polytope P, Eugène Ehrhart defined the function LP(t) := |tP ∩ Zm|,
which is the number of lattice points in the closed dilated polytope tP, for a nonnegative integer parameter t. Ehrhart showed that this function is a polynomial in t when P is
an integral polytope. More generally, if P is a rational polytope, the function LP(t) is
a quasi-polynomial whose period is closely related to the denominators appearing in the coordinates of the vertices of P [6, 2].
Liu and Osserman conjectured ([10, Conj. 4.2]) that polytopes associated to con-nected {1, 3}-graphs with the same number of vertices and edges have the same Ehrhart polynomial. They partially proved their conjecture, by showing that these quasi-polynomials coincide for all nonnegative odd values of the dilation parameter t. In 2013, Wakabayashi [15, Thm. A(ii)] proved their conjecture.
An ingredient in Wakabayashi’s proof for Liu and Osserman’s conjecture is a local transformation performed in {1, 3}-graphs. This transformation is called an A-move by Wakabayashi and it is also known as a nearest neighbor interchange (NNI). The NNI has been studied mainly for binary trees [4, 12], cubic graphs [14], and {1, 3}-graphs [15]. We present the following general result for connected graphs with the same degree sequence, which might of interest on its own. We refer to a vertex of degree one in a graph simply as a leaf. An edge is external if it is incident to a leaf, otherwise it is internal.
Theorem 1. Let G and G′
be connected graphs with the same degree sequence and the same set of external edges. Then
(a) G can be transformed into G′ through a series of NNI moves.
(b) One can choose a spanning tree in G and a spanning tree in G′ and require that all the pivots of the NNI moves are internal edges of both of these spanning trees.
In particular, for the special case of {1, 3}-graphs, the proof provided for Theorem 1(a) constitutes an alternative graph theoretic proof of a proposition by Wakabayashi [15, Prop. 6.2].
One of the concerns of this paper is on the scissors congruence conjecture for the uni-modular group, which is an analogue of Hilbert’s third problem (equidecomposability). Concretely, this was stated as the following question by Haase and McAllister [7]. An integral matrix U is unimodular if it has determinant ±1. An affine unimodular
trans-formation is defined by x → Ux + b, where U is a unimodular matrix and b is a real
Question 2. [7, Question 4.1] Suppose that P and P′ are polytopes with the same Ehrhart quasi-polynomial. Is it true that there is a decomposition of P into relatively open simplices
P1
, . . . , Pk and affine unimodular transformations U1
, . . . , Uk such that P′
is the disjoint union of U1
(P1
), . . . , Ur(Pr)?
We show that for polytopes associated to {1, 3}-graphs such a scissors congruence decomposition holds. Namely, we have the following.
Theorem 3. Let G and G′ be two connected {1, 3}-graphs with the same number of vertices and edges. Then there is a dissection of PG into smaller polytopes PG1, . . . , PGk
and affine unimodular transformations U1
, . . . , Uk such that P
G′ is the union of
U1
(Q1
G), . . . , Uk(QkG).
The proof of Theorem 3 relies on a piecewise unimodular transformation associated to a weighted version of the NNI move.
The rational polytope PG, associated to a {1, 3}-graph G, enjoys some fascinating
symmetry. Linke [9] considered the extension of LP(t) for all nonnegative real numbers t.
Royer [13] defined a polytope to be semi-reflexive if LP(s) = LP(⌊s⌋) for every nonnegative
real number s. One can verify that PG is semi-reflexive. A polytope is reflexive if it is
integral, the origin is in its interior, and it is semi-reflexive [2]. We prove the following.
Theorem 4. For each {1, 3}-graph G, the polytope 4PG−1 is reflexive.
The paper is organized as follows. Section 2 contains the results involving the NNI move in graphs with the same degree sequence, including Theorem 1, while Section 3 discusses the extension of the NNI move to weighted graphs. Section 4 describes the
unimodular decomposition of PG and presents the proof of Theorem 3. In Section 5, we
prove Theorem 4. Finally, Section 6 contains some concluding remarks.
2
Nearest neighbor interchange
With an eye towards Section 3, where we consider weighted graphs, we think of a graph as defined by Bondy and Murty [3]. A graph G is an ordered pair (V, E) consisting of a set V of vertices and a set E, disjoint from V , of edges, together with an incidence
function ψG that associates with each edge of G an unordered pair of (not necessarily distinct) vertices of G. The degree sequence of G is the monotonic nonincreasing sequence consisting of the degrees of its vertices.
A nearest neighbor interchange (NNI) is a local move performed in G on a trail W of length three. This move interchanges one end of the two extreme edges of W on the central edge (Figure 1). We refer to the central edge of W as the pivot of the NNI move. The
result of the move is another graph G′ on the same number of connected components,
vertices and edges, that is, G′ = (V, E), and only the incidence function ψ
G′ is adjusted accordingly. Intuitively, one can think of the edges as sticks that are being moved from G to G′
.
NNI
b b b b
a b a b
Figure 1: An NNI move on the trail marked in bold. One of the incidences of the edges a and b were interchanged.
We think of an NNI move as a function γW that associates the graph G to the graph G′.
In symbols, γW(G) = G′. Observe that W is also a trail in G′ and γW(G′) = G.
The well-known rotation, used in data structures to balance binary trees, is a particular NNI move, performed on a {1, 3}-tree (Figure 2).
left rotation right rotation r c b a r c b a b b b b
Figure 2: Right and left rotations applied to a binary tree.
An NNI move does not affect the incidence to leaves, thus it preserves the partition of the edge set into internal and external edges.
Culik and Wood [4, Thm. 2.4] proved that any two {1, 3}-trees with ℓ (labelled) leaves can be transformed into one another through a finite series of NNI moves. They
additionally gave an upper bound of 4ℓ − 12 + 4ℓ log2(ℓ/3) on the number of NNI moves
needed for this transformation. In this section, first we extend Culik and Wood’s theorem to trees with the same degree sequence (Lemma 8), which we then use to extend their result further to connected graphs with the same degree sequence (Theorem 1).
A caterpillar is a tree for which the removal of all leaves results in a path, called its
central path, or in the empty graph. For the later, we define that the central path is
empty.
Lemma 5. Any tree can be transformed into a caterpillar with the same degree sequence through a series of NNI moves.
Proof. Let T be a tree. If T is a caterpillar, there is nothing to prove. So, we may
would not be a longest path, as v is not in P . Let w and z be the two neighbors of u in P .
Let W be a trail with edges wu, uv, and vv′, where v′ is a neighbor of v other than u.
Perform an NNI on the trail W as in Figure 3, to insert v in P , obtaining another tree T′
with the same degree sequence and a path longer than P . By repeating this process, we obtain a desired caterpillar after a finite number of NNI moves.
v u w z w z v u w v u z b b b b b b b b b b b b b b b
Figure 3: An NNI to insert v in the central path.
The spine of a caterpillar is the sequence of the degrees of the vertices in the central path. We say the caterpillar is ordered if its spine is a monotonic nonincreasing sequence (Figure 4).
(a) 4b 5b 2b 3b (b) 5b 4b 3b 2b
Figure 4: (a) A caterpillar with spine (4, 5, 2, 3) and central path highlighted in bold. (b) An ordered caterpillar.
Lemma 6. Any caterpillar can be transformed into an ordered caterpillar with the same degree sequence through a series of NNI moves.
Proof. An NNI can be used to swap any two adjacent vertices in the central path of a
caterpillar, as in Figure 5(a). So we use an NNI to decrease, one by one, the number of inversions in the spine of a caterpillar until we obtain an ordered caterpillar.
Lemma 7. The external edges of a caterpillar can be sorted arbitrarily through a series of NNI moves.
Proof. An NNI can be used to swap any two external edges incident to two adjacent
vertices in the central path of a caterpillar, as in Figure 5(b).
z v u w z v u w v u z w (a) (b) d e a c b z v u w d c b a e z v u w a e d c b z v u w b b b b b b b b b b b b b b b b b b b b b b b b
Figure 5: (a) An NNI to swap two adjacent vertices in the central path of a caterpillar. (b) NNIs to swap labels on leaves hanging from adjacent vertices in the central path.
Lemma 8. Any two trees with the same degree sequence and the same set of external edges can be transformed into one another through a series of NNI moves.
Proof. Let T and T′ be two trees with the same degree sequence and the same set of
external edges. Using Lemmas 5 and 6, we obtain a series of NNI moves that transforms T into an ordered caterpillar with the same degree sequence and the same set of external
edges of T . Similarly, we obtain another series for T′. Using Lemma 7, we extend the
series of NNI moves for T to sort the external edges of the caterpillar coming from T
into the order they appear in the caterpillar coming from T′
. The composition of these
two series of NNI moves, with the series for T′ inverted, gives a series of NNI moves that
transforms T into T′.
Let G be a connected graph that is not a tree, and let e be an edge that is in a cycle.
The graph obtained from G by cutting e is the graph G′ resulting from the splitting of e
into two edges, each connecting one of the ends of e to one of two new leaves (Figure 6).
e G b b b b b b b b b b G′
Proof of Theorem 1. Let n be the number of vertices and m be the number edges in G
and G′. The proof of the theorem is by induction on the dimension r = m − n + 1 of the
cycle space of G and G′
. If r = 0, then G and G′
are trees and the theorem follows from Lemma 8. Hence we may assume that r > 0.
Choose an edge in a cycle of G and an edge in a cycle of G′. Let us denote these
two edges by e. Let H and H′
be the graphs obtained from G and G′
, respectively, by
cutting their edge e. The number of vertices in H and H′ is n + 2 and the number of
edges is m + 1. Call e′ and e′′ the two new edges in both H and H′. Since e is in a
cycle in G and in a cycle in G′, the graphs H and H′ are connected and the dimension
of their cycle space is r − 1. Also, H and H′ have the same degree sequence and the
same set of external edges. By induction, H can be transformed into H′ through a series
of NNI moves. The same series of NNI moves transforms G into G′. Indeed, the set of
external edges in all graphs obtained during the application of this series of NNI moves, transforming H into H′, contain e′ and e′′. Glue e′ and e′′ into an edge in each of these
graphs, obtaining a sequence of connected graphs that is the result of a series of NNI
moves starting at G and ending at G′. This ends the proof of (4).
G G
⇓cut ⇓cut
H = H0 NNI
←→ · · · ←→NNI Hk = H′
⇓glue ⇓glue ⇓glue
G = G0 NNI
←→ · · · ←→NNI Gk = G′
Similarly, by induction in the cycle space of G and G′
, one can prove (b). Indeed, it is sufficient to choose in each step of the induction an edge to be cut which is not in either of the spanning trees.
For the case of cubic graphs, the proof of Theorem 1 provides an alternative proof
for a theorem by Tsukui [14, Thm. II], which refers to NNI moves as ˜S-transformations,
where the ‘S’ stands for slide. For the slightly more general case of {1, 3}-graphs, the proof of Theorem 1 provides an alternative proof for a proposition by Wakabayashi [15, Prop. 6.2], which refers to NNI moves as A-moves. Wakabayashi’s proof uses a topological pants decomposition for compact, oriented surfaces of finite genus.
Figure 7 shows a series of NNI moves from the complete graph K4 to a tree with a
loop added to each of its leaves.
3
Weighted NNIs for
{1, 3}-graphs
To deal with weights on the edges of a {1, 3}-graph, we now enhance the NNI move. This was achieved by a bijection defined by Wakabayashi [15, Prop. 6.3].
b b b b b b b b b b b b b b b b b b b b b b b b +2 +1 −3 −2 +3 −1 +2 +1 −3 −2 +3 −1 +2 +1 +3 −3 −2 −1 +2 +1 −3 −2 +3 −1 +2 +1 −3 −2 +3 −1 +2 +1 −3 −2 +3 −1 NNI NNI NNI NNI NNI b b b b b b b b K4 T4
induced by a NNI move in G on W . The result of the move is the graph G′ obtained
from G by applying an NNI move on W , and the weight function w′ defined on the edges
of G′
as follows.
Let e be the central edge of W , that is, the pivot of the NNI move. Let a and b be the other edges in W , and c and d be the remaining edges adjacent to e, as depicted in
Figure 8. Possibly a, b, c, and d are not pairwise distinct. The weight function w′
is such that w′
f = wf for every f 6= e and
w′
e = we+ max{wa+ wc, wb+ wd} − max{wb+ wc, wa+ wd}. Note that, if w is integer valued, then so is w′
. Moreover, since pivots are always internal edges and an NNI move does not affect the partition of the edges into internal and external, a weighted NNI move may only change the weights of internal edges.
we wb wd wa wc w′ e wb wd wa wc b b b b G G′ wNNI
Figure 8: Edges and weights in a weighted nearest neighbor interchange.
We think of a weighted NNI move as a function φW(G, w) = (G′, w′), which extends
the previously defined NNI move, γW(G) = G′. Note that φW(G′, w′) = (G, w), because
γW(G′) = G and w′ e+ max{w ′ b+ w ′ c, w ′ a+ w ′ d} − max{w ′ a+ w ′ c, w ′ b+ w ′ d} = w′ e+ max{wb+ wc, wa+ wd} − max{wa+ wc, wb+ wd} = we+ max{wa+ wc, wb+ wd} − max{wb+ wc, wa+ wd} + max{wb+ wc, wa+ wd} − max{wa+ wc, wb+ wd} = we.
Wakabayashi [15, Prop. 6.3] proved that w ∈ tPG if and only if w′ ∈ tPG′ for every
integer t ≥ 0, which implies Liu and Osserman’s conjecture. We observe that this holds also for every real t ≥ 0, and state it below as we will use it in the next section.
Lemma 9. Let G be a {1, 3}-graph and w be a weight function defined on the edges of G. Let W be a trail in G of length three and suppose that φW(G, w) = (G′, w′). Then w ∈ tPG
if and only if w′ ∈ tP
G′ for every t ≥ 0.
Therefore there are distinct rational polytopes whose Ehrhart quasi-polynomials co-incide for all real t.
As a weighted NNI changes only the weight of the pivot, which is always an internal
In fact, by Theorem 1(b), one can choose a spanning tree T in G and a spanning tree T′
in G′ and require that the series of NNI moves uses as pivots only internal edges of T
and T′
. As a consequence, the bijection from PG to PG′ keeps fixed the majority of the
coordinates of the points. Namely, it changes only coordinates that correspond to internal edges of the chosen spanning trees. Formally, the latter discussion provides the following as a corollary of Theorem 1(b) and Lemma 9. Let E = {1, . . . , m} and w : E → R be a weight function defined on the edges of G. If X is a subset of E, then the restriction of
w to X is the function w|X : X → R such that w|X(x) = w(x) for all x ∈ X.
Corollary 10. Let G and G′ be connected {1, 3}-graphs with the same number of vertices and on the same set E of edges. Then there exists a bijection φ between PG and PG′ such
that, for w′ = φ(w), we have that
w′
|X = w|X
where E \ X is the set of internal edges of arbitrary spanning trees in G and G′.
4
Scissors congruence
Haase and McAllister [7] have raised Question 2 that can be thought of as an analogue of Hilbert’s third problem (equidecomposability) for the unimodular group. We show that for polytopes associated to {1, 3}-graphs a statement similar to Question 2 holds.
Let G be a {1, 3}-graph with edge set E = {1, . . . , m}. Let W be a trail in G
of length three and suppose that φW(G, w) = (G′, w′). As argued ahead, the function
φW(G, ·) : Rm → Rm is associated to one or two hyperplanes in Rm and two or four
unimodular transformations. For short, let φ(w) = φW(G, w) for every w ∈ Rm. Let a, b,
c, d, and e be as in Figure 8, with a, e, and b being the edges of W . Clearly φ is piecewise
linear, namely, for w ∈ Rm, w′
f = wf for every f 6= e and
w′ e= we+ wb− wd if wa+ wb ≥ wc+ wd and wa+ wd≥ wb+ wc, (1a) w′ e= we+ wa− wc if wa+ wb ≥ wc+ wd and wa+ wd< wb+ wc, (1b) w′ e= we+ wc− wa if wa+ wb < wc+ wd and wa+ wd≥ wb+ wc, (1c) w′ e= we+ wd− wb if wa+ wb < wc + wd and wa+ wd< wb+ wc. (1d)
The hyperplanes associated to φ are wa+ wb− wc − wd= 0 and wa− wb− wc + wd= 0,
which are either the same hyperplane (if a = b or c = d) or two orthogonal hyperplanes. Moreover, the matrix that gives the linear transformation in each case is unimodular. Indeed, the matrix for case (1a) is obtained from the identity matrix by substituting the
row corresponding to the edge e by the row χe + χb − χd, the matrix for case (1b) is
obtained from the identity matrix by substituting the row corresponding to the edge e
by the row χe+ χa− χc, the matrix for case (1c) is obtained from the identity matrix by
for case (1d) is obtained from the identity matrix by substituting the row corresponding
to the edge e by the row χe + χd − χb. Thus the determinant of each such matrix is
always 1. Therefore each of them is unimodular.
b b b b (0,1 2, 1 2) (1 2, 1 2, 0) (1 2, 0, 1 2) b b b b b b b (1 4, 1 4, 1 2) (1 2, 1 2, 0) b 1 2 3 w1 w2 w3 w1 w3 w2 b b w1+ w2+ w3 ≤ 1 2w1+ w3 ≤ 1 w1 ≤ w2+ w3 w3 ≤ 2w1 w2 ≤ w1+ w3 w3 ≥ 0 w3 ≤ w1+ w2 2w2+ w3 ≤ 1 w3 ≤ 2w2
Figure 9: The polytopes of the two cubic graphs on two vertices. The shaded triangles
are the intersection of the polytopes with the hyperplane w1 = w2.
Figure 9 shows an example with the two 3-regular graphs on two vertices and their
polytopes. The two graphs differ by one NNI move. The function φ(w) = w′ between the
two polytopes in Figure 9 is defined by w′
1 = w1, w2′ = w2, and w′ 3 = w3+ max{w1+ w2, w1+ w2} − max{2w1, 2w2} = w3+ w1+ w2− 2 max{w1, w2} = ( w3 − w1+ w2 if w1 ≥ w2 w3 + w1− w2 if w1 ≤ w2,
which is piecewise unimodular. In this case, only one hyperplane (the one containing the shaded triangle inside the polytopes, w1 = w2) splits the polytopes into two, each part
be-ing unimodularly equivalent to one of the parts of the other polytope. The correspondbe-ing unimodular transformations are given by
Uw1≤w2 = −1 0 −0 0 1 0 1 −1 1 Uw1≥w2 = 1 −0 −0 0 1 0 −1 1 1 .
polytopes may differ. Observe how the extra vertex (1 4,
1 4,
1
2) is “formed” when going from
the polytope on the left to the one on the right, and how it “disappears” when going
in the other direction. Also, the edge between the origin and the vertex (1
2, 1
2, 0) in the
polytope on the left is not an edge in the polytope on the right.
Theorem 3 extends this for graphs that differ by a series of NNI moves. It relates to Question 2.
Proof of Theorem 3. The proof is constructive. By Theorem 1, there exists a finite
se-quence of NNI moves that transforms G to G′, say G = G
0 ←→NNI · · · ←→NNI Gk = G′. We shall explain the procedure for the first two NNI moves assuming that both underlying weighted NNI moves are associated to only one hyperplane and consequently two uni-modular transformations. The other cases and the rest of the NNI moves follow in an analogous but more cumbersome fashion.
Let W0 be the trail used in the first NNI move, that goes from G = G0 to G1, and
let φ0(w) = φW0(G0, w), for w ∈ R
E, be the corresponding weighted NNI move. Let H
0be
the hyperplane associated to φ0, and let U1, U2 be the two unimodular transformations,
one for each side of the hyperplane H0. The hyperplane H0 dissects the polytope PG
into two smaller polytopes, Q1
G and Q
2
G, so that PG = Q1G ∪ Q
2
G. We now have that
PG1 = U1(Q
1
G) ∪ U2(Q2G). In words, we presented a dissection of PG into two smaller
polytopes and two affine unimodular transformations U1 and U2 that, if applied to the
two smaller polytopes, result in PG1. Now we will proceed one more step, and present a
dissection of PG into four smaller polytopes, and four affine unimodular transformations
that, if applied to the four smaller polytopes, will result in PG2 (Figure 10).
Let W1 be the trail used in the second NNI move, that goes from G1 to G2, and let
φ1(w) = φW1(G1, w), for w ∈ R
E, be the corresponding weighted NNI move. Let H
1 be
the hyperplane associated to φ1, and let T1, T2 be the two unimodular transformations,
one for each side of the hyperplane H1. The hyperplane H1 dissects the polytope PG1
into two smaller polytopes, Q1
G1 and Q 2 G1, so that PG2 = T1(Q 1 G1) ∪ T2(Q 2 G1).
Now, note that H1 dissects U1(Q1G) and U2(Q2G) each into two smaller polytopes,
obtaining the dissection PG1 = (Q 1 G1 ∩ U1(Q 1 G)) ∪ (Q 1 G1 ∩ U2(Q 2 G)) ∪ (Q 2 G1 ∩ U1(Q 1 G)) ∪ (Q 2 G1 ∩ U2(Q 2 G)). The latter naturally induces the following dissection:
b b b b b b (0,1 2, 0) (1 2, 0, 0) b (1 2, 0, 1 2) (1 2, 0, 0) b b b (1 4, 1 2, 1 4) (0, 0,1 2) NNI on W = (1, 2, 3) NNI on W = (1, 3, 2) b b b 1 3 2 w1 w2 w3 b w2 w3 w1 b b (1 2, 0, 1 2) (1 2, 1 2, 0) (0,1 2, 1 2) b b b b (1 4, 1 4, 1 2) (1 2, 1 2, 0) b 1 2 3 w1 w2 w3 bb w3 w2 w1 3 2 1 b (1 2, 1 2, 0) (1 4, 1 4, 1 2) b b b b (0,1 2, 1 2) (1 2, 1 2, 0) (1 2, 0, 1 2) b b w1 w3 w2 b w3 w2 w1 2 3 1 b b b NNI on W = (1, 3, 2) NNI on W = (1, 2, 3) (0, 0,1 2) (1 4, 1 2, 1 4) b b b (1 2, 0, 0) (1 2, 0, 1 2) b (1 2, 0, 0) (0,1 2, 0) b b b b b b 1 0 0 0 1 0 −1 −1 −1 Uw1≤w2 Uw1≥w2 1 0 0 0 1 0 −1 −1 −1 Uw1≥w3 1 0 0 −1 −1 −1 0 0 1 1 0 0 −1 −1 −1 0 0 1 Uw1≤w3
then PG = P11∪ P12∪ P21∪ P22, and PG2 = T1(Q 1 G1) ∪ T2(Q 2 G1) = T1(Q1 G1 ∩ U1(Q 1 G)) ∪ T1(Q 1 G1 ∩ U2(Q 2 G)) ∪T2(Q2 G1 ∩ U1(Q 1 G)) ∪ T2(Q 2 G1 ∩ U2(Q 2 G)) = T1U1(P11 ) ∪ T1U2(P12 )∪T2U1(P21 ) ∪ T2U2(P 22 ).
This completes the proof for the two first weighted NNI moves assuming that both are associated to only one hyperplane and consequently two unimodular transformations.
For the remaining cases, whenever a weighted NNI move is associated to two hy-perplanes (and consequently four unimodular transformations), the polytopes would be dissected into up to four smaller polytopes, but the process would be essentially the same.
5
Reflexivity
An equivalent way to define a reflexive polytope is to require that it is integral and has
the hyperplane description {x ∈ Rd | Ax ≤
1} for some integral matrix A. Another
equivalent definition of reflexivity is to say that a polytope P is reflexive if and only if the origin is in P◦ and (t + 1)P◦
∩ Zd = tP ∩ Zd for all t ∈ Z≥0, where P◦ is the interior
of P.
Proof of Theorem 4. The polytope 4PG−1 consists of the vectors w ∈ R
m satisfying
wa+ wb+ wc ≤ 1
wa− wb − wc ≤ 1
−wa+ wb − wc ≤ 1
−wa− wb+ wc ≤ 1,
for each degree three vertex v and edges a, b, and c incident to v, for a total of 4n3
inequal-ities, where n3 is the number of degree three vertices in G. From Liu and Osserman [10,
Prop. 3.5], the polytope 4PG is integral, and thus so is 4PG−1.
The reflexivity of the integral polytope 4PG−1 has some interesting consequences, as
follows. Let m denote the number of edges of G. Since 4PG is an m-dimensional integral
polytope, we have that for t ≡ 0 mod 4:
where h∗1, . . . , h∗
d−1, h∗d are the coefficients of the numerator of the rational function given
by the Ehrhart series of 4PG [2, Lemma 3.14].
The polytopes 4PG and 4PG−1 have the same Ehrhart polynomial. Hence it follows
from the reflexivity of 4PG−1 and from Hibi’s palindromic theorem [2, Thm. 4.6] that
h∗
k = h∗m−kfor all 0 ≤ k ≤ m/2. Therefore it follows that to describe L4PG(t), we only need
to compute half of its coefficients, thus lowering the computational complexity required to compute L4PG(t).
6
Concluding remarks and open problems
In the process of proving the main theorem, a great deal of structure of the polytopes PG,
and their corresponding cones has been revealed. The polytopes PG and their
corre-sponding cones may be related to the well-known metric polytopes and metric cones [5].
In particular, in the case that G is a planar graph, the polytope PG can be thought of as
a restricted type of “metric polytope” [8], associated to the dual graph G∗. It would be
interesting to pursue connections to metric polytopes.
Wakabayashi [15] gave a formula for two of the four constituent polynomials of the
Ehrhart quasi-polynomial for PG and for the volume of PG when G is a connected cubic
graph on n vertices. It is rather surprising that the Verlinde formula makes an appearance here, namely, Wakabayashi showed that, for odd t,
LPG(t) = (t + 2)n/2 2n+1 t+1 X j=1 1 sinnπj t+2 and vol(PG) = |Bn| 2 n!,
where Bn is the n-th Bernoulli number. Zagier [16] showed that the above Verlinde
formula is the polynomial
LPG(t) = (t + 2)n/2 2n+1 n/2 X k=0 (−1)k−122kB2 k (2k)! ck(t + 2) 2k,
where ck is the coefficient of x−2k in the Laurent expansion of sin−nx at x = 0. It would be interesting to determine if a similar formula holds for all connected {1, 3}-graphs. Even when G is a {1, 3}-tree, it is completely open, and quite interesting, to determine
the Ehrhart quasi-polynomial of PG. We determined the following quasi-polynomials for
G Ehrhart polynomial PG 1 24t 3 +1 4t 2 + ( 5 6t + 1, if t is even 11 24t + 1 4, if t is odd 1 240t 5 + 1 24t 4 + ( 5 24t 3 + 7 12t 2 +11 10t+ 1, if t is even 1 6t 3 +13t2+ 24079t+18, if t is odd 17 40320t 7 + 17 2880t 6 + n 144059 t 5 + 25 144t 4 +179 360t 3 +173 180t 2 +93 70t + 1, if t is even 103 2880t 5 + 35 288t 4 +1439 5760t 3 + 893 2880t 2 + 791 3360t + 1 16, if t is odd 31 725760t 9 + 31 40320t 8 + n 120960829 t 7 + 37 960t 6 + 653 4320t 5 +103 240t 4 +20413 22680t 3 +1723 1260t 2 +193 126t + 1, if t is even 43 6912t 7 +64019t 6 +345603181t 5 +123640t 4 +14515239205t 3 +403209923t 2 +2880379t + 1 32, if t is odd
Further generalizations to more general graphs and connections with other topics as manifold invariants would also be of interest.
7
Acknowledgements
We thank Tiago Royer and Fabrício Caluza Machado for valuable comments.
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